# 2.4 Descriptive statistics: box plot

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Box plots or box-whisker plots give a good graphical image of the concentration of the data. They also show how far from most of the data the extreme values are. The box plot is constructed from five values: the smallest value, the first quartile, the median, the third quartile, and the largest value. The median, the first quartile, and the third quartile will be discussed here, and then again in the section on measuring data in this chapter. We use these values to compare how close other data values are to them.

The median , a number, is a way of measuring the "center" of the data. You can think of the median as the "middle value," although it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median and half the values are the same number or larger. For example, consider the following data:

• 1
• 11.5
• 6
• 7.2
• 4
• 8
• 9
• 10
• 6.8
• 8.3
• 2
• 2
• 10
• 1

Ordered from smallest to largest:

• 1
• 1
• 2
• 2
• 4
• 6
• 6.8
• 7.2
• 8
• 8.3
• 9
• 10
• 10
• 11.5

The median is between the 7th value, 6.8, and the 8th value 7.2. To find the median, add the two values together and divide by 2.

$\frac{6.8+7.2}{2}=7$

The median is 7. Half of the values are smaller than 7 and half of the values are larger than 7.

Quartiles    are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile is the middle value of the lower half of the data and the third quartile is the middle value of the upper half of the data. To get the idea, consider the same data set shown above:

• 1
• 1
• 2
• 2
• 4
• 6
• 6.8
• 7.2
• 8
• 8.3
• 9
• 10
• 10
• 11.5

The median or second quartile is 7. The lower half of the data is 1, 1, 2, 2, 4, 6, 6.8. The middle value of the lower half is 2.

• 1
• 1
• 2
• 2
• 4
• 6
• 6.8

The number 2, which is part of the data, is the first quartile . One-fourth of the values are the same or less than 2 and three-fourths of the values are more than 2.

The upper half of the data is 7.2, 8, 8.3, 9, 10, 10, 11.5. The middle value of the upper half is 9.

• 7.2
• 8
• 8.3
• 9
• 10
• 10
• 11.5

The number 9, which is part of the data, is the third quartile . Three-fourths of the values are less than 9 and one-fourth of the values are more than 9.

To construct a box plot, use a horizontal number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box and the third quartile marks the other end of the box. The middle fifty percent of the data fall inside the box. The "whiskers" extend from the ends of the box to the smallest and largest data values. The box plot gives a good quick picture of the data.

You may encounter box and whisker plots that have dots marking outlier values. In those cases, the whiskers are not extending to the minimum and maximum values.

Consider the following data:

• 1
• 1
• 2
• 2
• 4
• 6
• 6.8
• 7.2
• 8
• 8.3
• 9
• 10
• 10
• 11.5

The first quartile is 2, the median is 7, and the third quartile is 9. The smallest value is 1 and the largest value is 11.5. The box plot is constructed as follows (see calculator instructions in the back of this book or on the TI web site ):

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